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Bitcoin is not illegal, but it is also not legally recognized by
governments as a currency. This gray area is what is leading
Apple to reject Bitcoin-transaction apps.
So even though Apple hasn't explicitly stated it won't allow
Bitcoin apps, its behavior certainly sends a clear message. This
is not to say that Apple will always reject Bitcoin
apps, but that Apple is simply protecting itself from legal
repercussions until regulators can come to some sort of consensus
on how to treat the digital currency.
This post from TechCrunch calls attention to how Apple has
thus far treated Bitcoin-related apps when they're submitted for
App Store approval. In short, not well.

Here are the salient details from the TechCrunch post:
AP
It seems like everyone wants to chime in with their opinions on
Bitcoin – "It's a scam," or "It's the future," etc. – but Apple
has so far maintained its famous silence when it comes to the
anonymous digital currency that's got heads turned across
industries.
Coinbase and Gliph are two higher-profile companies dealing in
Bitcoin – Coinbase is a Bitcoin exchange for buying and selling
the digital currency, and Gliph, a mobile payments app set up to
handle Bitcoin transactions. The Coinbase app was removed from
the App Store entirely and Apple requested that
Gliph get rid of its app's Bitcoin functionality, despite
letting it operate unencumbered for the previous seven months.
The rejection, [Gliph CEO Rob] Banagale notes, was based
largely off of section 22.1 of Apple’s App Store review
guidelines. The rule states that “apps must comply with all legal
requirements in any location where they are made available to
users. It is the developer’s obligation to understand and conform
to all local laws.”